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Theorem disji 5036
Description: Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑋 = 𝑌. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
disji.1 (𝑥 = 𝑋𝐵 = 𝐶)
disji.2 (𝑥 = 𝑌𝐵 = 𝐷)
Assertion
Ref Expression
disji ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐶𝑍𝐷)) → 𝑋 = 𝑌)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐵(𝑥)   𝑍(𝑥)

Proof of Theorem disji
StepHypRef Expression
1 inelcm 4379 . 2 ((𝑍𝐶𝑍𝐷) → (𝐶𝐷) ≠ ∅)
2 disji.1 . . . . . 6 (𝑥 = 𝑋𝐵 = 𝐶)
3 disji.2 . . . . . 6 (𝑥 = 𝑌𝐵 = 𝐷)
42, 3disji2 5035 . . . . 5 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
543expia 1123 . . . 4 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑌 → (𝐶𝐷) = ∅))
65necon1d 2962 . . 3 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐶𝐷) ≠ ∅ → 𝑋 = 𝑌))
763impia 1119 . 2 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝐶𝐷) ≠ ∅) → 𝑋 = 𝑌)
81, 7syl3an3 1167 1 ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐶𝑍𝐷)) → 𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  cin 3865  c0 4237  Disj wdisj 5018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-in 3873  df-nul 4238  df-disj 5019
This theorem is referenced by:  volfiniun  24444  fnpreimac  30728
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